The aim of this paper is to derive an energy stable finite element method (FEM) for simulating the dynamics of small droplets on solid, possibly non–homogeneous and inclined surfaces. The main challenges in the numerical framework for such problems come from the description of the evolving geometry. Surface tension, which plays an essential role in the mathematical model for droplet dynamics, is a function of the mean curvature which depends on the geometry of fluid–fluid interface. A recent review on the numerical models for surface tension is given in [1]}. Two essentially different approaches are widely used in the literature to describe the moving interface: implicit and explicit. In the implicit approach, a time–dependent scalar field is introduced on a fixed computational mesh to track the evolution of the interface. This is often referred to as the interface capturing approach and it is in a tight relationship with mesh adaptation methods. Indeed, mesh has to be sufficiently dense in the neighborhood of the interface in order for the position of the interface to be credibly captured on the discrete level. Level set method (see [2]} and references therein) and volume of fluid method (see [3]} and references therein) are representatives of the interface capturing approach. In interface tracking approach, the interface is described explicitly with an aligned mesh, i.e. the mesh fits the interface. Within this framework, if the interface moves, the mesh must also be moved with it accordingly. Lagrangian and arbitrary Lagrangian Eulerian (ALE) methods fall within the latter approach. Detailed description of the ALE methods can be found in [4]} while a concise review is given in Section  in this paper. Interface tracking methods are in general favored over interface capturing methods when a precise position of the interface is required. Their main disadvantage is the non–ability to capture possible topological changes in the domain.
